Unfortunately the presented method only works for the most simple linear recurrence relations.
Tangentially: look up the master theorem if you're interested in at least estimating growth rates for recurrence relations that crop up in computer science.
Just because of the impossibility of an exact formula for the roots of a high-degree polynomial that does not rule out the possibility of figuring out, say, the distribution of roots of such polynomials. The question is not about any polynomial in particular (hence Abel's theorem is no barrier).
edit: Think of the following example: take a polynomial a_n x^n + ... + a_0, where the coefficients a_i are i.i.d. Bernoulli random variables. Even though the degree n might be large (> 4) I can say with confidence that such a polynomial has a real root (x = 0) with probability 1/2. Similar though more sophisticated arguments are at work in the linked question.
This sounds like an organisational nightmare to be honest. You'd be going through the pile of exams multiple times (at least twice) and what do you do if there are multiple mistakes that are common in a single exam question?
Also: if you're sorting into "mistakes piles" for single exercises, how can you parallelise marking of separate and independent questions?
Even at top-notch universities (public or private), when I talk to retired faculty, grading almost always comes up as a reason they don't want to teach anymore.
> No grading is perfect, but there’s also some undercurrent of an attitude that students have paid to be there and are entitled to a certain grade.
Given that students have taken on hundreds of thousands of dollars in debt that they'll have to repay no matter what and on top of that a lot of jobs being completely out of reach these days without an academic degree (that for fucks sake isn't remotely required by virtually all jobs requiring it!), that's completely understandable.
Want to fix higher education? Bring the hammer down on companies abusing it as a proxy for legally discriminating against classes of society that are closely correlated with poor academic outcomes. Academic education should be reserved for the best of the best of our youth, and it should be fully paid for by the government, not simply another hurdle to pass to get a job that pays barely more than flipping burgers.
I think it is rational that students can feel entitled to that.
I also think that the vast majority of poorly paid, non-tenured professors and other teaching staff don't love being the targets of this harassment, since it's not their fault and largely out of their control, and it's not like they're getting the bulk of the tuition money. (That mostly goes to administrative expenses and sports programs.)
Heck, most adjunct faculty are often paid below minimum wage and qualify for food stamps.
> Given that students have taken on hundreds of thousands of dollars in debt that they'll have to repay no matter what and on top of that a lot of jobs being completely out of reach these days without an academic degree (that for fucks sake isn't remotely required by virtually all jobs requiring it!), that's completely understandable.
Would that my students were this engaged before the exam. Guess which students show up the most often for office hours? ... yeah, the ones that are getting the best grades.
If my students spent half as much time learning the subject as arguing with me about grades, they would be getting a higher grade than the one they are arguing for.
Online tools like Gradescope make this a little less painful (but still painful), but sometimes it's what's needed, especially on problems that are a little open-ended.
Regarding you last point: out of interest, what kind of venues were you thinking of? Be this personal blogs of said academics, just dumping it on a preprint server or actual ("formally published") publications?
Though often it is not implemented because it is quite complex (its details covering two thick books) and many of the special cases it covers rarely crop up in the real world, so the effort isn't worth it.
The caveat of Risch's algorithm is that it only "works" if the function you are trying to integrate has an elementary antiderivative. Many of the problems that Mathematica can solve (but SymPy fails at) involved special (i.e. non-elementary) functions.
That is definitely not true - I have an analog wrist watch (and am under 30). Also, there can be benefits to the use of analog clocks: in some circumstances they're quicker to read the time off of.
Have your circular dial divided in 24.658 sections then.
Nothing good can come from using a slightly longer second then what absolutely every piece of Earth hardware expects. Unless you are billion dollar rover mission where every bit or hardware is a custom design, it's imperative to use Earth seconds.
> I typically do a medium roast with no oily sheen to them and it serves me well.
I thought an oily/shiny appearance of coffee beans was due to them being roasted too long ago and then some of the coffee bean oils "leaking" out of the beans. Is this not the case, i.e. something that might even be controllable with the roasting process, even something desirable in some cases?
Oily beans are dark roasted and have a different flavour profile. If you get cheap supermarket espresso beans, they're usually roasted to this point. I think it gives a more consistent product given input variability, varietal differences are no longer so apparent.
I've only been roasting for a few months but the correlation seems to be with how dark you roast not with how long ago the roast happened. I roasted some beans a little over a week ago and they were covered in lots of oil one or two days later.
The difference is that in the Schrödinger case you're effectively 'turning' the solution (in the complex plane) which leads to the uncomfortable question of whether the solution to the heat equation you'd start with is still defined. When going from heat to Black-Scholes you're just rescaling in 'existing' dimensions which doesn't change the character of the PDE.
Tangentially: look up the master theorem if you're interested in at least estimating growth rates for recurrence relations that crop up in computer science.